An Analysis of Algebraic Codes over Lattice Valued Intuitionistic Fuzzy Type-3 R -Submodules

Abstract

In the last few decades, the algebraic coding theory found widespread applications in various disciplines due to its rich fascinating mathematical structure. Linear codes, the basic codes in coding theory, are significant in data transmission. In this article, the authors’ aim is to enlighten the reader about the role of linear codes in a fuzzy environment. Thus, the reader will be aware of linear codes over lattice valued intuitionistic fuzzy type-3 (LIF-3) R-submodule and $\alpha$ -intuitionistic fuzzy ( $\alpha$ -IF) submodule. The proof that the level set of LIF-3 is contained in the level set of $\alpha$ -IF is given, and it is exclusively employed to define linear codes over $\alpha$ -IF submodule. Further, $\alpha$ -IF cyclic codes are presented along with their fundamental properties. Finally, an application based on genetic code is presented, and it is found that the technique of defining codes over $\alpha$ -IF submodule is entirely applicable in this scenario. More specifically, a mapping from the ${\mathbb{Z}}_{64}$ module to a lattice $L$ (comprising 64 codons) is considered, and $\alpha$ -IF codes are defined along with the respective degrees.